Transcript Section 4.1

Math in Our World
Section 4.1
Early and Modern
Numeration Systems
Learning Objectives
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Define a numeration system.
Work with numbers in the Egyptian system.
Work with numbers in the Chinese system.
Identify place values in the Hindu-Arabic
system.
 Write Hindu-Arabic numbers in expanded
notation.
 Work with numbers in the Babylonian system.
 Work with Roman numerals.
Numeration Systems
A numeration system consists of a set of
symbols (numerals) to represent numbers,
and a set of rules for combining those
symbols.
A number is a concept, or an idea, used to
represent some quantity.
A numeral, on the other hand, is a symbol
used to represent a number.
Tally System
A tally system is the simplest kind of
numeration system, and almost certainly the
oldest.In a tally system there is only one
symbol needed and a number is represented
by repeating that symbol.
Most often, they are used to keep track of the number
of occurrences of some event. The most common
symbol used in tally systems is |, which we call a
stroke. Tallies are usually grouped by fives, with the fifth
stroke crossing the first four, as in ||||.
EXAMPLE 1
Using a Tally System
An amateur golfer gets the opportunity to play with
Tiger Woods, and, star struck, his game
completely falls apart. On the very first hole, it
takes him six shots to reach the green, then three
more to hole out. Use a tally system to represent
his total number of shots on that hole.
SOLUTION
The total number of shots is nine, which we tally up as
Simple Grouping Systems
In a simple grouping system there are symbols
that represent select numbers. Often, these
numbers are powers of 10. To write a number in
a simple grouping system, repeat the symbol
representing the appropriate value(s) until the
desired quantity is reached.
The Egyptian Numeration System
One of the earliest formal numeration systems was
developed by the Egyptians sometime prior to
3000 BCE. It used a system of hieroglyphics
using pictures to represent numbers.
EXAMPLE 2
Using the Egyptian
Numeration System
Find the numerical value of each Egyptian
numeral.
(a)
(b)
(c)
EXAMPLE 2
Using the Egyptian
Numeration System
SOLUTION
The value of any numeral is determined by counting up the
number of each symbol and multiplying the number of
occurrences by the corresponding value. Then the
amounts for each symbol are added.
(a)
= 10
=1
There are 4 heel bones and 3 staffs, so to find the value…
(4 x 10) + (3 x 1) = 40 + 3 = 43.
EXAMPLE 2
Using the Egyptian
Numeration System
SOLUTION
(b)
(3 x 100,000) + (3 x 10,000) + (2 x 100) + (3 x 10) + (6 x 1)
300,000 + 30,000 + 200 + 30 + 6 = 330,236
(c)
(1 x 1,000,000) + (2 x 10,000) + (2 x 1000) + (2 x 100) +
(1 x 10) + (3 x 1) = 1,000,000 + 20,000 + 2000 + 200 + 10 + 3
=1,022,213
EXAMPLE 3
Writing Numbers in Egyptian
Notation
Write each number as an Egyptian numeral.
(a) 42
(b) 3,200,419
EXAMPLE 3
Writing Numbers in Egyptian
Notation
SOLUTION
(a)Forty-two can be written as 4 x 10 + 2 x 1, so it consists of four
tens and two ones. We would write it using four of the tens symbol
(the heel bone) and two of the ones symbol (the vertical staff).
(b)Since 3,200,419 consists of 3 millions, 2 hundred thousands, 4
one hundreds, 1 ten, and 9 ones, it is written as
Positional Systems
In a positional system no multiplier is needed.
The value of the symbol is understood by its
position in the number. To represent a number
in a positional system you simply put the
numeral in an appropriate place in the number,
and its value is determined by its location.
Hindu-Arabic Numeration System
The numeration system we use today is called
the Hindu-Arabic system. It uses 10 symbols
called digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
This is a positional system since the position of
each digit indicates a specific value. The place
value of each number is given as
The number 82,653 means there are 8 ten thousands, 2
thousands, 6 hundreds, 5 tens, and 3 ones. We say that the
place value of the 6 in this numeral is hundreds.
EXAMPLE 9
Finding Place Values
In the number 153,946, what is the place value
of each digit?
a) 9
b) 3
c) 5
SOLUTION
(a) hundreds
(b) thousands
(c) ten thousands
(d) hundred thousands
(e) ones
d) 1
e) 6
Hindu-Arabic Numeration System
To clarify the place values, Hindu-Arabic
numbers are sometimes written in expanded
notation. An example, using the numeral
32,569, is shown below.
32,569 = 30,000 + 2,000 + 500 + 60 + 9
= 3 x 10,000 + 2 x 1,000 + 5 x 100 + 6 x 10 + 9
= 3 x 104 + 2 x 103 + 5 x 102 + 6 x 101 + 9
Since all of the place values in the Hindu-Arabic
system correspond to powers of 10, the system
is known as a base 10 system.
EXAMPLE 10
Writing a base 10 Number in
Expanded Form
Write 9,034,761 in expanded notation.
SOLUTION
9,034,761 can be written as
9,000,000 + 30,000 + 4,000 + 700 + 60 + 1
= 9 x 1,000,000 + 3 x 10,000 + 4 x 1,000 + 7 x 100 + 6 x 10 + 1
= 9 x 106 + 3 x 104 + 4 x 103 + 7 x 102 + 6 x 101 + 1.
Babylonian Numeration System
The Babylonians had a numerical system
consisting of two symbols. They are
and .
(These wedge-shaped symbols are known as
“cuneiform.”) The represents the number of
10s, and represents the number of 1s.
The ancient Babylonian system is sort of a
cross between a multiplier system and a
positional system.
EXAMPLE 11
Using the Babylonian
Numeration System
SOLUTION
Since there are 3 tens and 6 ones, the number
represents 36.
Babylonian Numeration System
You might think it would be cumbersome to write large
numbers in this system; however, the Babylonian
system was also positional in base 60.
Numbers from 1 to 59 were written using the two
symbols shown in Example 11, but after the number 60,
a space was left between the groups of numbers.
For example, the number 2,538 was written as
and means that there are 42 sixties and 18 ones. The
space separates the 60s from the ones.
EXAMPLE 12
Using the Babylonian
Numeration System
Write the numbers represented.
EXAMPLE 12
Using the Babylonian
Numeration System
SOLUTION
There are 52 sixties and 34 ones; so the number represents
52 x 60 = 3,120
+ 34 x 1 =
34
3,154
There are twelve 3,600s (602), fifty-one 60s and twenty-three
1s.
12 x 3,600 = 43,200
51 x
60 = 3,060
+ 23 x
1=
23
46,283
Roman Numeration System
The Romans used letters to represent their numbers.
The Roman system is similar to a simple grouping system,
but to save space, the Romans also used the concept of
subtraction. For example, 8 is written as VIII, but 9 is written
as IX, meaning that 1 is subtracted from 10 to get 9.
Roman Numeration System
There are three rules for writing numbers in Roman
numerals:
1. When a letter is repeated in sequence, its numerical value is
added. For example, XXX represents 10 + 10 + 10, or 30.
2. When smaller-value letters follow larger-value letters, the
numerical values of each are added. For example, LXVI
represents 50 + 10 + 5 + 1, or 66.
3. When a smaller-value letter precedes a larger-value letter,
the smaller value is subtracted from the larger value. For
example, IV represents 5 - 1, or 4, and XC represents 100 10, or 90.In addition, I can only precede V or X, X can only
precede L or C, and C can only precede D or M. Then 4 is
written as IV, 9 is written as IX, 40 is written as XL, 90 is
written XC, 400 is written as CD, and 900 is written as CM.
EXAMPLE 14
Using Roman Numerals
Find the value of each Roman Numeral.
(a) LXVIII
(b) XCIV
(d) CCCXLVI
(e) DCCCLV
(c) MCML
SOLUTION
(a) L = 50, X = 10, V = 5, and III = 3; so LXVIII = 68.
(b) XC = 90 and IV = 4; so XCIV = 94.
(c) M = 1,000, CM = 900, L = 50; so MCML = 1,950.
(d) CCC = 300, XL = 40, V = 5, and I = 1;
so CCCXLVI = 346.
(e) D = 500, CCC = 300, L = 50, V = 5; so DCCCLV = 855.
EXAMPLE 15
Writing Numbers Using
Roman Numerals
Write each number using Roman Numerals.
(a) 19
(b) 238
(c) 1,999
(d) 840
(e) 72
SOLUTION
(a) 19 is written as 10 + 9 or XIX.
(b) 238 is written as 200 + 30 + 8 or CCXXXVIII.
(c) 1,999 is written as 1,000 + 900 + 90 + 9 or MCMXCIX.
(d) 840 is written as 500 + 300 + 40 or DCCCXL.
(e) 72 is written as 50 + 20 + 2 or LXXII.